How can you solve a trapezoid problem without a graphing calculator?

[jesuslovesu]

Homework Statement

A gutter is to be made from a long strip of sheet metal 24 cm wide, by bending up equal amounts at each side through equal angles. Find the angle and the dimensions that will make the carrying capacity of the gutter as large as possible.

Homework Equations

The Attempt at a Solution

The equation I came up with is A = 24lsin$$\theta$$ - $$2l^2sin\theta$$ + $$.5*l^2sin(2\theta)$$

And when I take the partial derivatives I get two equations and set them to 0
24sin$$\theta$$ - 4lsin$$\theta$$ + 2lsin$$\theta$$cos$$\theta$$ = 0

24lcos$$\theta$$ - 2l^2cos$$\theta$$ + 2l^2cos$$(2\theta)$$ = 0

Substitution:
$$\frac{24^2*sin(\theta)}{4sin\theta-sin(\theta*2)} - \frac{2*(24sin\theta)^2}{(4sin\theta-sin(2\theta))^2}cos\theta + \frac{2(24sin(\theta))^2*cos(2\theta)}{4sin\theta - sin(2\theta)^2}$$

But how would one solve that without a graphing calculator? I can put it in and get 90 degrees (which I think is the correct answer). The book I'm using was written before calculators existed... I don't see any way to simplify that.

 

[NateTG]

Substitution:
$$\frac{24^2*sin(\theta)}{4sin\theta-sin(\theta*2)} - \frac{2*(24sin\theta)^2}{(4sin\theta-sin(2\theta))^2}cos\theta + \frac{2(24sin(\theta))^2*cos(2\theta)}{4sin\theta - sin(2\theta)^2}$$

But how would one solve that without a graphing calculator? I can put it in and get 90 degrees (which I think is the correct answer). The book I'm using was written before calculators existed... I don't see any way to simplify that.

$$\frac{24^2*sin(\theta)}{4sin\theta-sin(2\theta)} - \frac{2*(24sin\theta)^2}{(4sin\theta-sin(2\theta))^2}cos\theta + \frac{2(24sin(\theta))^2*cos(2\theta)}{(4sin\theta - sin(2\theta))^2}=0$$

Throw out some common factors and it's a bit easier to look at:
$$\sin(\theta) (4 \sin\theta - \sin(2 \theta))+ 2 \sin^2 \theta \cos \theta +2 \sin^2 \cos^2\theta$$
(Warning - this assumes that $(4sin\theta - sin(2\theta))^2 \neq 0$ -- you'll have to check whether that leads to an answer.)

Now, note that
$$2 \sin \theta \cos \theta = \sin 2\theta$$

$$4 \sin^2 \theta - \sin^2 \theta \cos \theta + 2 \sin^2 \theta \cos \theta + 2\sin^2 \theta \cos^2 \theta=0$$

Drop the $\sin^2$ since it's in all terms, and the rest is pretty straightforward.

P.S. You might want to follow this process to the end, and see what answer it leads to.